The finite gap method and the periodic NLS Cauchy problem of the anomalous waves, for a finite number of unstable modes

TL;DR

This paper applies the finite gap method to analyze the periodic NLS Cauchy problem with a finite number of unstable modes, revealing how initial data generate a sequence of interactions modeled by Akhmediev solitons, explaining rogue wave formation.

Contribution

It adapts the finite gap method to the NLS Cauchy problem with finite unstable modes, providing explicit elementary function solutions and insights into rogue wave dynamics.

Findings

Initial data partition time into intervals with varying unstable modes

Solution approximated by Akhmediev solitons within intervals

Reveals the role of finite N in rogue wave modeling

Abstract

The focusing Nonlinear Schr\"odinger (NLS) equation is the simplest universal model describing the modulation instability (MI) of quasi monochromatic waves in weakly nonlinear media, and MI is considered the main physical mechanism for the appearence of anomalous (rogue) waves (AWs) in nature. In this paper we study, using the finite gap method, the NLS Cauchy problem for generic periodic initial perturbations of the unstable background solution of NLS (what we call the Cauchy problem of the AWs), in the case of a finite number $N$ of unstable modes. We show how the finite gap method adapts to this specific Cauchy problem through three basic simplifications, allowing one to construct the solution, at the leading and relevant order, in terms of elementary functions of the initial data. More precisely, we show that, at the leading order, i) the initial data generate a partition of the time axis into a sequence of finite intervals, ii) in each interval $I$ of the partition, only a subset of ${\cal N}(I)\le N$ unstable modes are "visible", and iii) the NLS solution is approximated, for $t\in I$, by the ${\cal N}(I)$-soliton solution of Akhmediev type, describing the nonlinear interaction of these visible unstable modes, whose parameters are expressed in terms of the initial data through elementary functions. This result explains the relevance of the $m$-soliton solutions of Akhmediev type, with $m\le N$, in the generic periodic Cauchy problem of the AWs, in the case of a finite number $N$ of unstable modes.